I've a graph whose edges are weighted by probabilities, perhaps all equal. I would like to compute the overall probability of traveling between vertices x and y in the graph after I delete each edge vw with probability 1-p(vw). In other words, I'm trying to understand the probability that a self-avoiding walk goes from v to w. I cannot simply look at powers of the weighted adjacency matrix because this includes cycles.

I could imagine some recent-path-dependent ring-like object that when used in the adjacency matrix power operation eliminates counting short cycles. If the probabilities were low enough, this might produce a reasonable approximation with sane running time, although other properties of the graph might enter into the picture too.